Scale Drawings & Similar Figures

Similar figures have the same shape but different sizes. Matching angles are equal and matching sides are all multiplied by the same number, called the scale factor.

Find the scale factor by dividing a new side by the side it matches. Then every other side follows.

The surprise: doubling every length does not double the area, it quadruples it. A 2× model of a shape has 4× the area and 8× the volume. This is why a scale model of a building weighs far less than the scale would suggest.

Two figures are similar when there is a one-to-one correspondence of vertices with all corresponding angles congruent and all corresponding side lengths in a constant ratio k. Triangle similarity can be established with less: AA is sufficient, as are SAS and SSS.

Under a similarity with ratio k, any length (side, perimeter, altitude, radius) scales by k, any area scales by k², and any volume by k³. The exponent is the dimension of the measurement, since area is a product of two lengths and volume of three.

This dimensional scaling shows up well beyond geometry class. It explains why a scale model cannot be strong in proportion to the original — cross-sectional area (strength) grows as k² while volume (weight) grows as k³ — and it is the reason surface-area-to-volume arguments appear in biology and engineering.